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Difference between revisions of "2003 AMC 12A Problems/Problem 24"
 
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== Problem ==
 
== Problem ==
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If <math>a\geq b > 1,</math> what is the largest possible value of <math>\log_{a}(a/b) + \log_{b}(b/a)?</math>
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Objects <math>A</math> and <math>B</math> move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object <math>A</math> starts at <math>(0,0)</math> and each of its steps is either right or up, both equally likely. Object <math>B</math> starts at <math>(5,7)</math> and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
 
  
 
<math>
 
<math>
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\mathrm{(A)} \ 0.10 \qquad
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\mathrm{(A)}\ -2      \qquad
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\mathrm{(B)} \ 0.15 \qquad
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\mathrm{(B)}\ 0     \qquad
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\mathrm{(C)} \ 0.20 \qquad
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\mathrm{(C)}\ 2      \qquad
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\mathrm{(D)} \ 0.25 \qquad
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\mathrm{(D)}\ 3      \qquad
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\mathrm{(E)} \ 0.30 \qquad
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\mathrm{(E)}\ 4
 
</math>
 
</math>
  
 
== Solution ==
 
== Solution ==
 +
Using logarithmic rules, we see that
 +
 +
<cmath>\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)</cmath>
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<cmath>=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)</cmath>
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Since <math>a</math> and <math>b</math> are both greater than <math>1</math>, using [[AM-GM]] gives that the term in parentheses must be at least <math>2</math>, so the largest possible values is <math>2-2=0 \Rightarrow \boxed{\textbf{B}}.</math>
  
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If <math>A</math> and <math>B</math> meet, their paths connect <math>(0,0)</math> and <math>(5,7).</math> There are <math>\binom{12}{5}=792</math> such paths, so the probability is <math>\frac{792}{2^{6}\cdot 2^{6}} \approx 0.20 \Rightarrow \boxed{C}</math>
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Note that the maximum occurs when <math>a=b</math>.
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==Solution 2 (More Algebraic)==
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Similar to the previous solution, we use our logarithmic rules and start off the same way.
 +
<cmath>\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)</cmath>
 +
 
 +
However, now, we use a different logarithmic rule stating that <math>\log_{a}b</math> is simply equal to <math>\frac {\log_{10}b}{\log_{10}a}</math>. With this, we can rewrite our previous equation to give us <cmath>2-(\log_{a}b+\log_{b}a) = 2 - \left(\frac {\log_{10}b}{\log_{10}a} + \frac {\log_{10}a}{\log_{10}b}\right)</cmath>.
 +
 
 +
We can now cross multiply to get that <cmath>2 - \left(\frac {\log_{10}b}{\log_{10}a} + \frac {\log_{10}a}{\log_{10}b}\right) = 2 - \left(\frac {2 \cdot \log_{10}b \cdot \log_{10}a}{\log_{10}b \cdot \log_{10}a}\right)</cmath> Finally, we cancel to get <math>2-2=0 \Rightarrow \boxed{\textbf{(B) 0}}.</math>
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Solution by: armang32324
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==Solution 3 (Calculus)==
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By the logarithmic rules, we have <math>2-(\log_a{b}+\frac{1}{\log_a{b}})</math>.
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Let <math>x=\log_a{b}</math>. Thus, the expression becomes <math>2-(x+\frac{1}{x})</math>. We want to find the maximum of the function. To do so, we will find its derivative and let it equal to 0. We get: <math>\frac{d}{dx}\big(2-(x+\frac{1}{x})\big)=\frac{d}{dx}\big(2-x-\frac{1}{x})=-1+x^{-2}=0 \implies \frac{1}{x^2}=1, x^2=1, x=\pm1.</math>
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Since <math>a\geq b>1, x=-1</math> is not a solution. Thus, <math>x=1</math>. Substituting it into the original expression <math>2-(x+\frac{1}{x})</math>, we get <math>2-(1+\frac{1}{1})=2-2=\boxed{0}</math>.
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==Video Solution==
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The Link: https://www.youtube.com/watch?v=InF2phZZi2A&t=1s
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-MistyMathMusic
  
 
== See Also ==
 
== See Also ==
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{{AMC12 box|year=2003|ab=A|num-b=23|num-a=25}}
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{{AMC12 box|year=2003|ab=A|num-b=23|after=25}}
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{{MAA Notice}}

Latest revision as of 08:35, 6 September 2023

Problem

If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$

$\mathrm{(A)}\ -2 \qquad \mathrm{(B)}\ 0 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 3 \qquad \mathrm{(E)}\ 4$

Solution

Using logarithmic rules, we see that

\[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\] \[=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)\]

Since $a$ and $b$ are both greater than $1$, using AM-GM gives that the term in parentheses must be at least $2$, so the largest possible values is $2-2=0 \Rightarrow \boxed{\textbf{B}}.$

Note that the maximum occurs when $a=b$.


Solution 2 (More Algebraic)

Similar to the previous solution, we use our logarithmic rules and start off the same way. \[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\]

However, now, we use a different logarithmic rule stating that $\log_{a}b$ is simply equal to $\frac {\log_{10}b}{\log_{10}a}$. With this, we can rewrite our previous equation to give us \[2-(\log_{a}b+\log_{b}a) = 2 - \left(\frac {\log_{10}b}{\log_{10}a} + \frac {\log_{10}a}{\log_{10}b}\right)\].

We can now cross multiply to get that \[2 - \left(\frac {\log_{10}b}{\log_{10}a} + \frac {\log_{10}a}{\log_{10}b}\right) = 2 - \left(\frac {2 \cdot \log_{10}b \cdot \log_{10}a}{\log_{10}b \cdot \log_{10}a}\right)\] Finally, we cancel to get $2-2=0 \Rightarrow \boxed{\textbf{(B) 0}}.$

Solution by: armang32324

Solution 3 (Calculus)

By the logarithmic rules, we have $2-(\log_a{b}+\frac{1}{\log_a{b}})$. Let $x=\log_a{b}$. Thus, the expression becomes $2-(x+\frac{1}{x})$. We want to find the maximum of the function. To do so, we will find its derivative and let it equal to 0. We get: $\frac{d}{dx}\big(2-(x+\frac{1}{x})\big)=\frac{d}{dx}\big(2-x-\frac{1}{x})=-1+x^{-2}=0 \implies \frac{1}{x^2}=1, x^2=1, x=\pm1.$ Since $a\geq b>1, x=-1$ is not a solution. Thus, $x=1$. Substituting it into the original expression $2-(x+\frac{1}{x})$, we get $2-(1+\frac{1}{1})=2-2=\boxed{0}$.

Video Solution

The Link: https://www.youtube.com/watch?v=InF2phZZi2A&t=1s

-MistyMathMusic

See Also

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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All AMC 12 Problems and Solutions

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